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no. 5
One-sided convergence in the Boltzmann–Grad limit
Thierry Bodineau1; Isabelle Gallagher2; Laure Saint-Raymond3; Sergio Simonella3
1 CMAP, Ecole Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau Cedex, France
2 DMA, École normale supérieure, CNRS, PSL Research University, 75005 Paris, France and UFR de mathématiques, Université Paris-Diderot, Sorbonne Paris-Cité, 75013 Paris, France
3 ENS de Lyon, Université de Lyon, UMPA UMR 5669 CNRS, 46 allée d’Italie, 69364 Lyon Cedex 07, France
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 5, pp. 985-1022.

We review various contributions on the fundamental work of Lanford [21] deriving the Boltzmann equation from (reversible) hard-sphere dynamics in the low density limit.

We focus especially on the assumptions made on the initial data and on how they encode irreversibility. The impossibility to reverse time in the Boltzmann equation (expressed for instance by Boltzmann’s H-theorem) is related to the lack of convergence of higher order marginals on some singular sets. Explicit counterexamples single out the sets with vanishing measure where the initial data should converge in order to produce the Boltzmann dynamics.

Ce papier présente diverses contributions basées sur le travail fondamental de Lanford [21] qui a permis d’obtenir l’équation de Boltzmann à partir de la dynamique (réversible) des sphères dures dans la limite de densité faible.

On s’intéresse en particulier aux hypothèses sur la donnée initiale et sur la façon dont elles codent l’irréversibilité. On montre que l’impossibilité de renverser le sens du temps dans l’équation de Boltzmann (qui est exprimée notamment dans le théorème H) est liée à l’absence de convergence des marginales d’ordre supérieur sur des ensembles singuliers. Un contre exemple explicite permet de caractériser les ensembles, de mesure asymptotiquement nulle, où la donnée initiale doit converger pour obtenir la dynamique de Boltzmann.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1589

Thierry Bodineau 1; Isabelle Gallagher 2; Laure Saint-Raymond 3; Sergio Simonella 3

1 CMAP, Ecole Polytechnique, CNRS, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau Cedex, France
2 DMA, École normale supérieure, CNRS, PSL Research University, 75005 Paris, France and UFR de mathématiques, Université Paris-Diderot, Sorbonne Paris-Cité, 75013 Paris, France
3 ENS de Lyon, Université de Lyon, UMPA UMR 5669 CNRS, 46 allée d’Italie, 69364 Lyon Cedex 07, France
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
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Thierry Bodineau; Isabelle Gallagher; Laure Saint-Raymond; Sergio Simonella. One-sided convergence in the Boltzmann–Grad limit. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 27 (2018) no. 5, pp. 985-1022. doi : 10.5802/afst.1589. https://afst.centre-mersenne.org/articles/10.5802/afst.1589/

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